How to add different physics properties to different edges of a same object?
I'm afraid that what you are tying to do will not give the desired result.
You could for instance select you sphere, go to the Properties window and under the Physics Properties tab click Soft Body. In the Soft Body settings there is a section called Edges where you can change values related to the behavior of edges in soft body physics.
These values however apply to all the edges of the mesh of your object, and thus the mesh as a whole. It does not allow for giving different values to individual edges.
Disconnecting edges from each-other would also result in the the object failing apart under influence of physics simulation.
(Also: The term 'Edge Split' refers to making visually hard edges on a smooth shaded object. An edge between to meeting faces (that looked smooth using shading) could be made to look sharp. Because smooth shading applies to faces, it does not work on a mesh without faces.)